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| + | [[File:Pentro.png|thumb|297px|Visualization of boogafive using exponentiation]] |
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| − | '''Boogafive''' is equal to 5↑↑↑5, or 5 [[pentation|pentated]] to 5. The name is formed by adding the [[booga-]] prefix to 5. |
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| + | '''Boogafive''' is equal to \(5\uparrow\uparrow\uparrow5\), or 5 [[pentation|pentated]] to 5.<ref>{{cite web|first=Sbiis|last=Saibian|url=https://sites.google.com/site/largenumbers/home/3-2/2|title=The Fz, The Fuga & The Megafuga|accessdate=2017-03-10}}</ref> |
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| + | Googology Wiki user [[User:Hyp cos|Hyp cos]] calls this number '''pentro''', and it's equal to s(5,5,3) = s(5,2,4) = s(5,3,1,2) in [[strong array notation]].<ref>[https://stepstowardinfinity.wordpress.com/2015/07/24/lan-numbers/ Numbers from linear array notation | Steps Toward Infinity!]</ref> |
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| + | |||
| + | == Etymology == |
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| + | The name "boogafive" is formed by adding the [[booga-]] prefix to 5. |
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| + | |||
| + | The name "pentro" is based on the word "penta" and "tri" (Greek for 5 and 3), meaning a base-5 number, generated using hyper-3 operation based on exponentiation. |
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| + | |||
| + | == Computation == |
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| + | Boogafive can be computed in the following process: |
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| + | *\(f(1)=5\) |
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| + | *\(f(2)=5^5=3,125\) |
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| + | *\(f(3)=5^{5^5}\approx1.911\times10^{2,184}\) |
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| + | *\(f(4)=5^{5^{5^5}}\) |
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| + | *etc. |
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| + | *Boogafive is equal to \(f(f(f(f(5))))\). |
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| + | |||
| + | == Approximations == |
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| + | {{approx| |
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| + | |up=\(5\uparrow\uparrow\uparrow5\) (exact) |
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| + | |chain=\(5\rightarrow5\rightarrow3\) (exact) |
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| + | |beaf=\(\{5,5,3\}\) (exact) |
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| + | |steinhaus=5[5] |
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| + | |hypere=\(E(5)1\#1\#5\) (exact) |
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| + | |hyperf=\(6!2\) |
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| + | |fast=\(f_4(4)\) |
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| + | |hardy=\(H_{\omega^4}(4)\) |
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| + | |slow=\(g_{\zeta_0}(5)\) |
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| + | }} |
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| + | |||
| + | == Sources == |
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| + | <references /> |
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| + | == See also == |
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{{Fuga- family}} |
{{Fuga- family}} |
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| + | {{Hyp cos' linear strong array notation numbers}} |
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[[Category:Numbers]] |
[[Category:Numbers]] |
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[[Category:Up-arrow notation level]] |
[[Category:Up-arrow notation level]] |
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| + | [[Category:3 entry linear]] |
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| + | [[Category:Powers of 5]] |
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| + | [[Category:Perfect powers with exponent coprime to 6]] |
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| + | [[Category:Pentation]] |
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| + | [[Category:Booga-]] |
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Revision as of 04:50, 12 July 2021
Visualization of boogafive using exponentiation
Boogafive is equal to \(5\uparrow\uparrow\uparrow5\), or 5 pentated to 5.[1]
Googology Wiki user Hyp cos calls this number pentro, and it's equal to s(5,5,3) = s(5,2,4) = s(5,3,1,2) in strong array notation.[2]
Etymology
The name "boogafive" is formed by adding the booga- prefix to 5.
The name "pentro" is based on the word "penta" and "tri" (Greek for 5 and 3), meaning a base-5 number, generated using hyper-3 operation based on exponentiation.
Computation
Boogafive can be computed in the following process:
- \(f(1)=5\)
- \(f(2)=5^5=3,125\)
- \(f(3)=5^{5^5}\approx1.911\times10^{2,184}\)
- \(f(4)=5^{5^{5^5}}\)
- etc.
- Boogafive is equal to \(f(f(f(f(5))))\).
Approximations
| Notation | Approximation |
|---|---|
| Arrow notation | \(5\uparrow\uparrow\uparrow5\) (exact) |
| Bowers' Exploding Array Function | \(\{5,5,3\}\) (exact) |
| Chained arrow notation | \(5\rightarrow5\rightarrow3\) (exact) |
| Hyper-E notation | \(E(5)1\#1\#5\) (exact) |
| Hyperfactorial array notation | \(6!2\) |
| Steinhaus-Moser Notation | 5[5] |
| Fast-growing hierarchy | \(f_4(4)\) |
| Hardy hierarchy | \(H_{\omega^4}(4)\) |
| Slow-growing hierarchy | \(g_{\zeta_0}(5)\) |
Sources
- ↑ Saibian, Sbiis. The Fz, The Fuga & The Megafuga. Retrieved 2017-03-10.
- ↑ Numbers from linear array notation | Steps Toward Infinity!
See also
Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex
Tetentri group: duprimitol · duprimitolplex · duprimibol · tetentri
Tetentet group: truprimitol · truprimitolplex · truprimibol · truprimibolplex · tetentet
Tetenpent group: quadprimitol · quadprimitolplex · quadprimibol · quadprimibolplex · tetenpent
Tetenhex group: quinprimitol · quinprimitolplex · quinprimibol · quinprimibolplex · tetenhex
Chainprimol group: chainprimol · chainprimolplex · chainpribol · chainpribolplex · chainduprimol · chainduprimolplex · chaindupribol · chaindupribolplex · chaintruprimol · chaintruprimolplex · chaintrupribol · chaintrupribolplex
Pententri group: duchainol · duchainolplex · duchainbol · duchainbolplex · duchainprimol · duchainprimolplex · duchainpribol · duchainpribolplex · duchainduprimol · duchainduprimolplex · duchaindupribol · pententri
Pententet group: truchainol · truchainolplex · truchainbol · truchainbolplex · truchainprimol · truchainprimolplex · truchainpribol · truchainpribolplex · truchainduprimol · truchainduprimolplex · truchaindupribol · truchaindupribolplex · pententet · pentenpent · pentenhex
Choichainol group: choichainol · choichainolplex · choichainbolplex · choichainpribolplex · choichaindupribolplex · choiduchaindupribolplex
Hexentri group: hexentri · hexentet · hexenpent · hexenhex